Ema drew quadrilateral ABCD. If side AB = 40y, side BC = 4+3x, side CD = 8y+32, and side AD = 9x-20, for what values of x and y can Ema be sure that her quadrilateral is a parallelogram?(1 point)

Question

Responses

x = 40 and y = 16
x = 40 and y = 16

x = 20 and y = 8
x = 20 and y = 8

x = 1 and y = 4
x = 1 and y = 4

x = 4 and y = 1

Answer 1

To check if quadrilateral ABCD is a parallelogram, we need to ensure that the opposite sides are equal. This gives us the following equations:

1. \( AB = CD \)
2. \( BC = AD \)

Substituting the given values into these equations, we have:

1. \( 40y = 8y + 32 \)
2. \( 4 + 3x = 9x - 20 \)

Now, let’s solve both equations.

Solving Equation 1:

\[ 40y = 8y + 32 \] Subtract \( 8y \) from both sides: \[ 40y - 8y = 32 \] This simplifies to: \[ 32y = 32 \] Divide both sides by 32: \[ y = 1 \]

Solving Equation 2:

\[ 4 + 3x = 9x - 20 \] Subtract \( 3x \) from both sides: \[ 4 = 6x - 20 \] Add 20 to both sides: \[ 24 = 6x \] Divide both sides by 6: \[ x = 4 \]

Thus, we find: \[ x = 4 \quad and \quad y = 1 \]

Now, let’s check back to see if this pair of values makes both pairs of opposite sides equal:

1. For AB and CD:

   • \( AB = 40y = 40(1) = 40 \)
   • \( CD = 8y + 32 = 8(1) + 32 = 8 + 32 = 40 \) (they are equal)

2. For BC and AD:

   • \( BC = 4 + 3x = 4 + 3(4) = 4 + 12 = 16 \)
   • \( AD = 9x - 20 = 9(4) - 20 = 36 - 20 = 16 \) (they are equal)

Since both pairs of opposite sides are equal, it confirms that quadrilateral ABCD is a parallelogram when \( x = 4 \) and \( y = 1 \).

Therefore, the correct answer is: x = 4 and y = 1.